L(s) = 1 | + 2-s + (−0.618 + 1.61i)3-s + 4-s + (−0.618 + 1.61i)6-s + (−0.381 − 2.61i)7-s + 8-s + (−2.23 − 2.00i)9-s + 4.47i·11-s + (−0.618 + 1.61i)12-s − 1.23·13-s + (−0.381 − 2.61i)14-s + 16-s + 5.23i·17-s + (−2.23 − 2.00i)18-s + 8.47i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.356 + 0.934i)3-s + 0.5·4-s + (−0.252 + 0.660i)6-s + (−0.144 − 0.989i)7-s + 0.353·8-s + (−0.745 − 0.666i)9-s + 1.34i·11-s + (−0.178 + 0.467i)12-s − 0.342·13-s + (−0.102 − 0.699i)14-s + 0.250·16-s + 1.26i·17-s + (−0.527 − 0.471i)18-s + 1.94i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.852522346\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.852522346\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.618 - 1.61i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.381 + 2.61i)T \) |
good | 11 | \( 1 - 4.47iT - 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 5.23iT - 17T^{2} \) |
| 19 | \( 1 - 8.47iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 7.70iT - 29T^{2} \) |
| 31 | \( 1 - 2.76iT - 31T^{2} \) |
| 37 | \( 1 + 0.763iT - 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 4.94iT - 43T^{2} \) |
| 47 | \( 1 + 6.47iT - 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 7.23iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 7.23iT - 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.43818702649946889995170431497, −9.662706753134410956310718751659, −8.521067081240456403604668978034, −7.43433655877233605711032841261, −6.68441058385257951394634051310, −5.68336357185931872345313926735, −4.81262740963840833702965431464, −4.02328040981591173493777339452, −3.36536578159568296727130980861, −1.67404563871674716494633397110,
0.69110077047701441249159540905, 2.52524051686233231605210669586, 2.92452096857446816300009630787, 4.69365316308581384196775364594, 5.46030595039366272031243137357, 6.19742712720476688517660351696, 6.95072063394678907984108244673, 7.84343890121371246165101792915, 8.784866465680478567129044671031, 9.524808744883528074728109374798